(2) If \(\mathcal {X}=\mathcal {Y}=\mathcal {C}\) , then \(\xi \)  is the class of split triangles by Theorem REF . So \((\mathcal {C}, \mathbb {E}_\xi , \mathfrak {s}_\xi )\)  is an exact category by [1]}. Conversely, we assume that \((\mathcal {C}, \mathbb {E}_\xi , \mathfrak {s}_\xi )\)  is an exact category. Then any triangle \((T2):@C=2em{ A[r]^f&B[r]^g&C[r]^{h\ \ \ \ }&\Sigma A}\)  in \(\xi \)  satisfies that \(f\)  is monomorphic. Thus the triangle \((T2)\)  is split, and hence \(\xi \)  is the class of split triangles. So \(\mathcal {X}=\mathcal {Y}=\mathcal {C}\) . This completes the proof.\(\Box \)

